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  "body": "\n## Definition\n\nLet $k$ be a [[field|04C5ZBG1WT]] and let $\\mathcal{T}$ be a $k$-linear [[triangulated category|T2V7GJJ2EN]] with shift $X \\mapsto X[1]$.\nAn object $E$ of $\\mathcal{T}$ is *exceptional* if $\\Hom(E, E) = k$ and $\\Hom(E, E[l]) = 0$ for every integer $l \\ne 0$.\nAn *exceptional collection* is a sequence $(E_1, \\dots, E_n)$ of objects of $\\mathcal{T}$ such that:\n\n1. each $E_i$ is exceptional;\n2. $\\Hom(E_j, E_i[l]) = 0$ for all $j \u003e i$ and all integers $l$.\n\nThe collection is *full* if the smallest triangulated subcategory of $\\mathcal{T}$ that contains $E_1, \\dots, E_n$ and is closed under isomorphisms is $\\mathcal{T}$ itself.\nIt is *strong* if moreover $\\Hom(E_i, E_j[l]) = 0$ for all $i, j$ and all $l \\ne 0$.\n\n## Remarks\n\nExample: on the projective space $\\mathbb{P}^n$ over $k$, the line bundles $(\\mathcal{O}, \\mathcal{O}(1), \\dots, \\mathcal{O}(n))$ form a full strong exceptional collection.\nIt lives in the [[bounded derived category of coherent sheaves|2VP2D51KB8]], and this is a theorem of Beilinson.\nClause 2 holds since $\\Hom(\\mathcal{O}(j), \\mathcal{O}(i)[l]) = H^l(\\mathbb{P}^n, \\mathcal{O}(i - j))$, and for $-n \\le i - j \u003c 0$ all cohomology of $\\mathcal{O}(i - j)$ vanishes.\nNon-example: on an elliptic curve $C$ over $k$, no line bundle $L$ is exceptional, since $\\Hom(L, L[1]) = H^1(C, \\mathcal{O}_C) \\cong k \\ne 0$.\nA full exceptional collection gives a semiorthogonal decomposition $\\mathcal{T} = \\langle \\mathcal{A}_1, \\dots, \\mathcal{A}_n \\rangle$, where $\\mathcal{A}_i$ is the subcategory generated by $E_i$, equivalent to the bounded derived category of finite-dimensional vector spaces over $k$.\n",
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